Conceptual portion of Relativistic Jets Problem

[Darth Tader]

1. The full problem statement, all variables and given/known data

The problem can be found http://www.astro.umd.edu/~chris/Teaching/ASTR498E_Spring_2006/homework5.pdfif any of my description is lacking.

I have solved successfully part a), most of b), and c) for the problem, but part d) asks:

The best evidence that AGN jets contain material moving at relativistic velocities comes from observations of “superluminal motion”.

(b) Show that, for a given velocity v, the apparent velocity Vapp is maximized when cosθ = v/c
and, at that maximum, has a value Vapp = Γv where Γ = (1 − v²/c²)^-1/2 is the standard Lorentz factor.

(c) In the M87 jet, we see blobs with apparent motions of Vapp ≈ 6c. Calculate an approximate
value for the velocity of these blobs.

(d) Observations of some systems reveal blob motions in both the approaching jet and the receding jet (i.e. the counter-jet). Blobs in the receding jet are always seen to be moving
slower than those in the approaching jet, and never attain apparent superluminal velocities.
If we assume that both jets have the same true velocity and are oriented back-to-back, show
that one can solve uniquely for both the jet speed v and jet angle θ if you know the apparent
speeds of blobs in both jets. [Hint — you might find it useful to consider drawing lines on
the (v,θ)-plane.]

Homework Equations

Through parts a-c), I have determined that the transverse velocity β_T = $\frac{sin(theta)}{1-βcos(theta)}$ and that the angle which maximizes this equation is equal to β which is equal to v/c.

My answer to part c) is 2.96 x 10⁸ m/s.

The Attempt at a Solution

b): I am unclear on how to do this portion of the problem, but I believe it to be a simple algebra step I am missing, so I do not expect help on this section.

c): Unless I am reading this question incorrectly, I believe that one can see if the jets are back to back and their true velocities are equivalent that the angle θ (angle from observer's plane t and the jet's actual velocity) must be equal by simple geometry. Is this assumption naive? Or do I need to draw a triangle similarly to how I did in part a) and solving for the velocity as I did throughout the problem parts above?

Thanks for your help, it is greatly appreciated!

 

[anathema]

Thank you for sharing your progress on this problem. It seems like you have a good understanding of the problem so far. Here are some suggestions for part d:

1. Start by drawing a diagram that shows the orientation of the jets, the observer's plane, and the apparent velocities of the blobs in both jets.

2. From the diagram, you can see that the apparent velocities of the blobs in both jets are related to the true velocity of the jets and the angle θ.

3. Use the formula you found in part b to relate the apparent velocities to the true velocity and θ.

4. Since the jets are back to back, the angle θ will be the same for both jets. This means that you will have two equations with two unknowns (v and θ).

5. Solve the system of equations to find the values of v and θ.

I hope this helps. Good luck with your problem!